Sampling from complex, unnormalized probability densities is a fundamental challenge in Bayesian inference and probabilistic modeling. While Markov chain Monte Carlo (MCMC) methods provide asymptotic guarantees, they often suffer from slow mixing and high computational costs due to fixed or manually tuned trajectory lengths. In this work, we propose a novel framework that treats trajectory termination as a learnable component of the sampling dynamics. By framing MCMC within the theory of non-acyclic generative flow networks (GFlowNets), we train state-dependent neural classifiers to decide when a trajectory has reached a high-density region and should terminate. We theoretically establish the connection between optimal classifiers and the target density via detailed balance conditions and introduce a multilevel training scheme to facilitate exploration in complex geometries. Experimental results across various benchmark densities demonstrate that our approach significantly reduces average trajectory lengths while improving mode coverage and mixing compared to standard MCMC baselines.
Amortized inference is the task of training a parametric model, such as a neural network, to approximate a distribution with a given unnormalized density where exact sampling is intractable. When sampling is implemented as a sequential decision-making process, reinforcement learning (RL) methods, such as generative flow networks, can be used to train the sampling policy. Off-policy RL training facilitates the discovery of diverse, high-reward candidates, but existing methods still face challenges in efficient exploration. We propose to use an adaptive training distribution (the \teacher) to guide the training of the primary amortized sampler (the \student). The \teacher, an auxiliary behavior model, is trained to sample high-loss regions of the \student and can generalize across unexplored modes, thereby enhancing mode coverage by providing an efficient training curriculum. We validate the effectiveness of this approach in a synthetic environment designed to present an exploration challenge, two diffusion-based sampling tasks, and four biochemical discovery tasks demonstrating its ability to improve sample efficiency and mode coverage. Source code is available at https://github.com/alstn12088/adaptive-teacher.
Sampling-based methods offer a principled approach to uncertainty quantification in Bayesian neural networks. Their practical use, however, is often challenged by the computational cost of exploring high-dimensional and multimodal posterior distributions. To overcome these difficulties, Bayesian Deep Ensembles, i.e., warmstarting the sampling from several optimized solutions, have proven to be an effective strategy. In this paper, we demonstrate that curvature estimates computed during the warmstart as a byproduct in adaptive optimizers such as AdamW can inform the sampling phase at negligible additional cost. Specifically, our proposed preconditioned sampling strategy based on optimizer-derived geometries can substantially reduce or even eliminate the need for a lengthy sampling burn-in phase and leads to greater numerical stability. This approach consistently maintains or improves predictive performance and uncertainty quantification without any additional computational costs. We confirm the consistency of our findings across various datasets and network architectures.
Moritz Schlager, Emanuel Sommer, Thomas Möllenhoff +1
Sampling from unnormalized densities is analogous to the generative modeling problem, but the target distribution is defined by a known energy function instead of data samples. Because evaluating the energy function is often costly, a primary challenge is to learn an efficient sampler. We introduce Flow Sampling, a framework built on diffusion models and flow matching for the data-free setting. Our training objective is conditioned on a noise sample and regresses onto a denoising diffusion drift constructed from the energy function. In contrast, diffusion models' objective is conditioned on a data sample and regresses onto a noising diffusion drift. We utilize the interpolant process to minimize the number of energy function evaluations during training, resulting in an efficient and scalable method for sampling unnormalized densities. Furthermore, our formulation naturally extends to Riemannian manifolds, enabling diffusion-based sampling in geometries beyond Euclidean space. We derive a closed-form formula for the conditional drift on constant curvature manifolds, including hyperspheres and hyperbolic spaces. We evaluate Flow Sampling on synthetic energy benchmarks, small peptides, large-scale amortized molecular conformer generation, and distributions supported on the sphere, demonstrating strong empirical performance.